advertisement

Eight Versions of Bayes’s Theorem Name Formu la Com ments This is the simplest form, derivable from the conjunction rule. When H Ö E, it reduces to P(H|E) = P(H) / P(E), an important special case Simple form P(H|E) = P(H) P(E|H) P(E) Explicit form P(H|E) = P(H) P(E|H) P(H) P(E|H) + P(~H) P(E|~H) This is obtained from the simple form by expanding P(E) according to the theorem on total probability (TTP) General form P(H 1|E) = P(H1) P (E |H 1) P(H 1) P (E |H 1) + ... + P(H n) P E|H n) This is again obtained from the simple form by the TTP on the assumptio n that H 1, ..., H n form a partition Sigma form P(H 1|E) = Canceled form P(H|E) = P(H 1) P (E |H 1) Gi P(H i) P (E |H i) 1 1+ Odd s form Relative odds form Compound odds form P(~H) P(E|~H) × P(H) P(E|H) P(H|E) P(H) = P(~H|E) P(~H) P(H 1|E) = P(H 2|E) P(H 1) P(H 2) This is a compact way of writing the general form, of which it is a notational variant This is obtained from the explicit form by dividing the numerator and the denominator by P(H)P(E|H) and collecting terms. It is very useful when we replace E with the testimony of multiple independent witnesses × P(E|H) P(E|~H) This is useful because, as in the canceled form, the term P(E) has dropp ed out. The likelihood ratio on the right is the “Bayes factor,” the term by which the prior odds are multiplied to obtain the posterior odds × P(E |H 1) P(E |H 2) This generalization of the o dds form is useful when we do not have a partition but we wish to know the relative impact of E on two hyp otheses. P(H|A&B ) P(H) P(A|H) P(B|H) = × × P(~H|A&B) P(~H) P(A|~H) P(B|~H) This form allows combination of evidence, assuming that ±H screens off A from B. W ithout this assumption, the probabilities in the final term would have to be conditional on A as well. Given this assumption, we can combine evidence by multiplying the prio r odds by successive B ayes factors. Copyright © Tim McGrew 2005